Shape optimization method for strength design problem of microstructures in a multiscale structure

Abstract

In this paper, we propose a solution to a shape optimization problem for the strength design of periodic microstructures in multiscale structures. Two maximum stress minimization problems are addressed; minimization of the maximum stress of the microstructure and minimization of the maximum stress of the macrostructure. The homogenization method is used to bridge the macrostructure and the microstructures, and also to calculate the local stress in the microstructures. By replacing the maximum value of the stress with a Kreisselmeier-Steinhauser function, the difficulty of non-differentiability on the maximum stress is avoided. Each strength design problem is formulated as a distributed parameter optimization problem with the area constraint including the whole microstructures. The shape gradient functions for both problems are derived using Lagrange's undetermined multiplier method, the material derivative method, and the adjoint variable method. The H1 gradient method is used to determine the unit cell shapes of the microstructures, while reducing the objective function and maintaining the smooth design boundaries. In the numerical examples, the optimal shapes obtained for the maximum local stress minimization of the microstructure and the macrostructure are compared and discussed. The results confirm the effectiveness of the microstructure shape optimization method for the two strength design problems of multiscale structures.